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Convergence analysis of an adaptive nonconforming finite element method

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Abstract

An adaptive nonconforming finite element method is developed and analyzed that provides an error reduction due to the refinement process and thus guarantees convergence of the nonconforming finite element approximations. The analysis is carried out for the lowest order Crouzeix-Raviart elements and leads to the linear convergence of an appropriate adaptive nonconforming finite element algorithm with respect to the number of refinement levels. Important tools in the convergence proof are a discrete local efficiency and a quasi-orthogonality property. The proof does neither require regularity of the solution nor uses duality arguments. As a consequence on the data control, no particular mesh design has to be monitored.

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References

  1. Agouzal, A.: A posteriori error estimator for nonconforming finite element methods. Appl. Math. Lett. 7, 1017–1033 (1994)

    Article  MathSciNet  Google Scholar 

  2. Ainsworth, M.: Robust a posteriori error estimation for non-conforming finite element approximation. SIAM J Numer. Anal. 42 (2005) no 6, 2320–2341.

    Article  MATH  MathSciNet  Google Scholar 

  3. Ainsworth, M., Oden, J.T.: A posteriori error estimation in finite element analysis. Wiley, Chichester, 2000

  4. Babuska, I., Strouboulis, T.: The finite element method and its reliability. Clarendon Press, Oxford, 2001

  5. Bahriawati, C., Carstensen, C.: Three Matlab implementations of the lowest-order Raviart-Thomas MFEM with a posteriori error control. Computational Methods in Applied Mathematics, Vol 5 (2005) No 4, 333–361

    MATH  MathSciNet  Google Scholar 

  6. Bangerth, W., Rannacher, R.: Adaptive finite element methods for differential equations. Lectures in Mathematics. ETH-Zürich. Birkhäuser, Basel, 2003

  7. Binev, P., Dahmen, W., DeVore, R.: Adaptive finite element methods with convergence rates. Numer. Math. 97, 219–268 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Braess, D.: Finite elements. Cambridge University Press, Cambridge, 1997

  9. Braess, D., Verfürth, R.: Multigrid methods for nonconforming finite element methods. SIAM J. Numer. Anal. 27, 979–986 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  10. Brenner, S.: An optimal order multigrid method for nonconforming finite elements. Math. Comp. 52, 1–15 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  11. Brenner, S., Scott, L.R.: The mathematical theoty of finite element methods. Springer, New York, 1994

  12. Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. Springer, Berlin-Heidelberg-New York, 1991

  13. Carstensen, C.: Quasi-interpolation and a posteriori error analysis in finite element method. M2AN 33, 1187–1202 (1999)

    Google Scholar 

  14. Carstensen, C., Bartels, S.: Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. I. Low order conforming, nonconforming and mixed FEM. Math. Comp. 71(239), 945–969 (2002)

    MathSciNet  Google Scholar 

  15. Carstensen, C., Bartels, S., Jansche, S.: A posteriori error estimates for nonconforming finite element methods. Numer. Math. 92, 233–256 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Carstensen, C., Bolte, J.: Adaptive mesh-refining algorithm for courant finite elements in matlab. data structures and algorithms. (2006) (in preparation)

  17. Carstensen, C., Hoppe, R.H.W.: Error reduction and convergence for an adaptive mixed finite element method. To appear in Math. Comp. (2006)

  18. Ciarlet, P.G.: The finite element method for elliptic problems. Nort-Holland, Amsterdam, 1978

  19. Dari, E., Duran, R., Padra, C., Vampa, V.: A posteriori error estimators for nonconforming finite element methods. RAIRO Model. Math. Anal. Numer. 30, 385–400 (1996)

    MATH  MathSciNet  Google Scholar 

  20. Dörfler, W.: A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. 33(3), 1106–1124 (1996)

    Article  Google Scholar 

  21. Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Computational differential equations. Cambridge University Press, Cambridge, 1995

  22. Hoppe, R.H.W., Wohlmuth, B.: Adaptive multilevel iterative techniques for nonconforming finite element discretizations. East-West J. Numer. Math. 3, 179–197 (1995)

    MATH  MathSciNet  Google Scholar 

  23. Hoppe, R.H.W., Wohlmuth, B.: Element-oriented and edge-oriented local error estimators for nonconforming finite element methods. RAIRO Model. Math. Anal. Numer 30, 237–263 (1996)

    MATH  MathSciNet  Google Scholar 

  24. Kanschat, G., Suttmeier, F.-T.: A posteriori error estimates for nonconforming finite element schemes. Calcolo 36, 129–141 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  25. Marini, L.D.: An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method. SIAM J. Numer. Anal. 22(3), 493–496 (1985)

    Article  MathSciNet  Google Scholar 

  26. Morin, P., Nochetto, R.H., Siebert, K.G.: Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38(2), 466–488 (2000)

    Article  MathSciNet  Google Scholar 

  27. Morin, P., Nochetto, R.H., Siebert, K.G.: Local problems on stars: a posteriori error estimators, convergence, and performance. Math. Comp. 72(243), 1067–1097 (2003)

    Article  MathSciNet  Google Scholar 

  28. Oswald, P.: On a hierarchical basis multilevel method with nonconforming P1 elements. Numer. Math. 62, 189–212 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  29. Schieweck, F.: A posteriori error estimates with post-processing for nonconforming finite elements. ESAIM Math. Mod. Numer. Anal. 36, 489–503 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  30. Verfürth, R.: A review of a~posteriori error estimation and adaptive mesh-refinement techniques. Wiley-Teubner, New York, Stuttgart, 1996

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Authors and Affiliations

  1. Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Carsten Carstensen

  2. Institute of Mathematics, Universität Augsburg, 86159, Augsburg, Germany

    Ronald H.W. Hoppe

  3. Department of Mathematics, University of Houston, Houston, TX, 77204-3008, USA

    Ronald H.W. Hoppe

Authors
  1. Carsten Carstensen
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  2. Ronald H.W. Hoppe
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Supported by the DFG Research Center MATHEON ``Mathematics for key technologies'' in Berlin.

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Carstensen, C., Hoppe, R. Convergence analysis of an adaptive nonconforming finite element method. Numer. Math. 103, 251–266 (2006). https://doi.org/10.1007/s00211-005-0658-6

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  • DOI: https://doi.org/10.1007/s00211-005-0658-6

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